54 
ME. A. CAYLEY ON THE EQUATION EOE A EUNCTION 
Avhence changing the sign of d, 
a^0^-{-9(ac-hy-^ -27 D ; 
or multiplying the two equations and putting for 
u{a^u-\-9i{ac—lf)Y-\-27 □ =n3{ti— (a— 13)*}, 
that is, the equation of differences is 
a%®+18(ac— 5^)aV+81(«c— □ =0 ; 
but this mode of composition of the equation of differences is peculiar to the case of the 
cubic. 
If in the several equations in 6 we substitute for the seminvariants the covariants to 
which they respectively belong, we obtain as follows : — 
22. For the cubic equation («, h, c, d\v, 1)®=0, the equation for ^^=(/3 — y)(^— “i/)) is 
1)^=0. 
U’x 
9x 
-v/— 27n X 
A 
+ 1 
0 
H 
r ^ 
+ 1 
23. For the quartic equation (a, h, c, d, eyjv, 1)^=0, the equation for 
^(=(/^-y)(y— is 
U®x 
K 
512 U □ X 
256 □ X 
+ 1 
-3JU 
+ 2IH 
+ 1 
1 )^= 0 . 
24. And for the quintic equation (a, h, c, d, e.fjv, 1)®=0, the equation for 
a{={|3-y)(/3-W-0(5'-8X5'-0(s-s)(*-«y)*)« 
f U'^ • ^ 
0 , 
625U®(A, B, C, D, E, F, y)\ 
^ 12500x/dU^{4U(No. 14)^-U(No. 20)-50(No. 15)(No. 16)}, (S, 1)^=0, 
15625 □ {-3U^(No. 14)+25(No. 15)^}, 
.76125 □x/U 
where the covariant which enters into the coefficient of being of the sixth degree in 
the coefficients, is not given in the Tables. 
Its value (completed for me, from the first term, by Mr. Davis) is 
